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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 6 — Mar. 17, 2008
  • pp: 3865–3870
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Elliptical discrete solitons supported by enhanced photorefractive anisotropy

Peng Zhang, Jianlin Zhao, Fajun Xiao, Cibo Lou, Jingjun Xu, and Zhigang Chen  »View Author Affiliations


Optics Express, Vol. 16, Issue 6, pp. 3865-3870 (2008)
http://dx.doi.org/10.1364/OE.16.003865


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Abstract

We demonstrate elliptical discrete solitons in an optically induced two-dimensional photonic lattice. The ellipticity of the discrete soliton results from enhanced photorefractive anisotropy and nonlocality under a nonconventional bias condition. We show that the ellipticity and orientation of the discrete solitons can be altered by changing the direction of the lattice beam and/or the bias field relative to the crystalline c-axis. Our experimental results are in good agreement with the theoretical prediction.

© 2008 Optical Society of America

1. Introduction

Light propagation in periodic photonic structures exhibits many intriguing phenomena. Examples in the linear regime include transmission spectrum band-gaps, Bloch modes, and diffraction management. In the nonlinear regime, a typical example is the self-localized wave packets (discrete or gap solitons), which have been a subject of intense investigation [1

1. For a review, see D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003). [CrossRef] [PubMed]

]. Much of the work has been done in optically induced photonic lattices in photorefractive crystals [2–6

2. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E 66, 046602 (2002). [CrossRef]

]. Since the photorefractive nonlinearity is intrinsically anisotropic and nonlocal [7

7. A. A. Zozulya and D. Z. Anderson, “Propagation of an optical beam in a photorefractive medium in the presence of a photogalvanic nonlinearity or an externally applied electric field,” Phys. Rev. A. 51, 1520–1531 (1995). [CrossRef] [PubMed]

], a nature question arises: how would the anisotropy and nonlocality influence the self-localized states in induced photonic lattices, or in nonlinear discrete systems in general? In a continuum photorefractive system, both theoretical and experimental studies have shown that coherent and incoherent elliptical solitons can arise from either anisotropic nonlinearity or anisotropic coherence (correlation function) [8–10

8. A. A. Zozulya, D. Z. Anderson, A. V. Mamaev, and M. Saffman, “Self-focusing and soliton formation in media with anisotropic nonlocal material response,” Europhys. Lett. 36, 419–424 (1996). [CrossRef]

]. However, such issues have received little attention in a discrete system [11–16

11. O. Katz, T. Carmon, T. Schwartz, M. Segev, and D. N. Christodoulides, “Observation of elliptic incoherent spatial solitons,” Opt. Lett. 29, 1248–1250 (2004). [CrossRef] [PubMed]

]. Recently, theory has shown that anisotropic linear diffraction and moving elliptical discrete solitons exist in two-dimensional waveguide lattices induced in photorefractive media based on an isotropic model [12

12. J. Hudock, N. K. Efremidis, and D. N. Christodoulides, “Anisotropic diffraction and elliptic discrete solitons in two-dimensional waveguide arrays,” Opt. Lett. 29, 268–270 (2004). [CrossRef] [PubMed]

, 13

13. F. Ye, L. Dong, J. Wang, T. Cai, and Y. Li, “Discrete elliptic solitons in two-dimensional waveguide arrays,” Chin. Opt. Lett. 3, 227–229 (2005).

]. Although an anisotropic model has been utilized to analyze the formation of discrete solitons [14–16

14. P. G. Kevrekidis, D. J. Frantzeskakis, R. Carretero-González, B. A. Malomed, and A. R. Bishop “Discrete solitons and vortices on anisotropic lattices,” Phys. Rev. E72, 046613 (2005). [CrossRef]

], the elliptical discrete solitons possessing obvious ellipticity were never observed in experiment. In this paper, we predict in theory and demonstrate in experiment a novel class of elliptical discrete solitons. We show by employing an anisotropic photorefractive model that the soliton ellipticity is originated from enhanced photorefractive anisotropy and nonlocality under a nonconventional bias condition.

2. The theoretical model

We first propose a model, considering both the photorefractive anisotropy and different orientation configurations, to describe the propagation of a probe beam in an optically induced photonic lattice under a nonconventional bias field. A coordinates system is constructed with placing the x axis along one of the principal axes of the lattice beam as shown in Fig. 1. The angles of the external bias field and the crystalline c-axis with respect to the x axis are denoted by θe and θc, respectively. The probe and lattice beam propagate collinearly along the z axis. Here, the polarization directions of the probe and lattice beam are always kept to be parallel and perpendicular to the c-axis, respectively. Then the dimensionless equations governing the steady state propagation of the probe beam in the induced lattice can be written as [2

2. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E 66, 046602 (2002). [CrossRef]

, 4–5

4. D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Królikowski, “Spatial solitons in optically induced gratings” Opt. Lett. 28, 710–712 (2003). [CrossRef] [PubMed]

, 7

7. A. A. Zozulya and D. Z. Anderson, “Propagation of an optical beam in a photorefractive medium in the presence of a photogalvanic nonlinearity or an externally applied electric field,” Phys. Rev. A. 51, 1520–1531 (1995). [CrossRef] [PubMed]

, 9

9. P. Zhang, J. Zhao, C. Lou, X. Tan, Y. Gao, Q. Liu, D. Yang, J. Xu, and Z. Chen, “Elliptical solitons in nonconventionally biased photorefractive crystals,” Opt. Express 15, 536–544 (2007). [CrossRef] [PubMed]

, 17

17. P. Zhang, Y. Ma, J. Zhao, D. Yang, and H. Xu, “One-dimensional spatial dark soliton-induced channel waveguides in lithium niobate crystal,” Appl. Opt. 45, 2273–2278 (2006). [CrossRef] [PubMed]

]

Fig. 1. Geometry of the relative orientation of the crystalline c-axis, bias field, and lattice beam.
(zi22)B(r)=i(φxcosθc+φysinθc)B(r)
(1a)
2φ+φ·ln(1+I)=E0[ln(1+I)xcosθe+ln(1+I)ysinθe]
(1b)
I=V(x,y)+B(r)2
(1c)

The solitary solutions of Eqs. (1) can be found in the form B(x, y, z)=b(x, y)exp(iβz), where β is the propagation constant, and the real envelope b(x, y) satisfies the following equation

(β122)b(x,y)=(φxcosθc+φysinθc)b(x,y)
(2)

where φ is also determined by Eqs. (1b) and (1c).

3. Variation of induced lattice structures

Fig. 2. Refractive index distributions of optically induced photonic lattices under different conditions, where in each figure the green and purple arrow indicate the directions of the crystalline c-axis and bias field, respectively, and the number in the upper left corner corresponds to the index modulation depth at E 0 = 1.

4. Numerical results of discrete diffraction and self-trapping

Now we study the linear and nonlinear propagation dynamics in the induced lattices in the presence of enhanced nonlocality and anisotropy by use of both beam propagation method and the iteration method. Let B(x,y)=0.6exp[(x2+y2)64] , E 0 = 1, Λ = 8, and z = 110, the output intensity pattern of the probe beam (for on-site excitation) after linear propagation is depicted in Fig. 3(a), where the panels from left to right correspond to the cases of θe = θc = π/4; θe = 3π/4, θc = π/4; θe = 5π/8, θc = π/8; and θe = π/2, θc = 0, respectively. Notice that, for comparison, we include one conventional bias case, i.e., θe = θc = π/4. In the presence of nonlinearity, the input Gaussian beams can evolve into self-trapped states as shown in Figs. 3(b). It is clear that different conditions will lead to different patterns of linear diffractions and nonlinear self-trapped states. The soliton solutions with various peak intensities obtained by Petviashvili iteration method [18

18. V. I. Petviashvili, “Equation of an extraordinary soliton,” Sov. J. Plasma Phys. 2, 257–258 (1976).

] for each case are displayed in Figs. 3(c) and 3(d), which shows that the solitons with higher peak intensities exhibit stronger localization. By comparing Figs. 3(b) and 3(c), we can see that the soliton profiles at lower peak intensities obtained by the nonlinear beam propagation and iteration method are very similar. Note that in the presence of photorefractive anisotropy, even under conventional bias condition, the soliton profile is somewhat elliptical (see the left column in Fig. 3). With enhanced anisotropy and nonlocality under E 0c, the ellipticity of the solitons will be dramatically increased, and the orientations of elliptical solitons can be altered by changing the relative orientations of lattice beam, bias direction, and the c-axis. For the case with θe = π/2 and θc = 0, the equivalent lattice spacing is reduced while the index modulation is relatively low [see Fig. 2(c), top], thus the coupling between waveguide channels is much stronger, and both linear diffraction and nonlinear self-trapped state resembles that in the continuum limit (not well discretized).

Fig. 3. Numerical results for discrete diffraction and self-trapped states. (a)–(b) are the output patterns of the probe beam under linear (a) and nonlinear (b) propagation obtained by beam propagation method. Animations show the corresponding evolution dynamics. (c)–(d) show soliton solutions at different peak intensities obtained with iteration method. The columns from left to right are for the cases of θe = θc = π/4; θe = 3π/4, θc = π/4; θe = 5π/8, θc = π/8; and θe = π/2, θc = 0, respectively. [Media 1][Media 2]

5. Experimental results of discrete diffraction and self-trapping

To perform experimental demonstrations, we use the experimental setup similar to that used in Ref. [5

5. H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, “Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices,” Phys. Rev. Lett. 92, 123902 (2004). [CrossRef] [PubMed]

]. The photonic lattices are created by sending a partially coherent beam through an amplitude optical mask, and a focused Gaussian laser beam as a probe beam propagates collinearly with the lattice beam. The probe beam and the lattice beam have the same wavelength of 532nm, and their polarization directions are kept to be parallel and perpendicular to the c-axis, respectively. A photorefractive SBN:60 crystal (with dimensions of 6.5mm×6.5mm×6.7(c)mm and 0.025% by weight chromium dopant) is used. The intensity ratio between the probe and lattice beam is adjusted to be about 3:1. In our experiment, the lattice spacing is about 23µm. We first block off the probe beam until the lattice structure arrive steady-state, after that the linear and nonlinear transport of the probe beam is monitored simply by taking its instantaneous (before nonlinear self-action) and steady-state (after self-action) output patterns after propagating through the lattice. Because the induced lattice potential in our experiment is relatively shallow due to the weak absorption of the crystal sample at 532nm, thus a simulation using the experimental parameters is also performed for comparison.

6. Summary

In summary, we have demonstrated a novel class of elliptical discrete solitons in a nonconventionally biased photorefractive crystal. The ellipticity of the discrete solitons is originated from an enhanced photorefractive anisotropy and nonlocality. It is revealed that both the bias direction and the orientation of the lattice beam can dramatically influence the structure and modulation depth of the lattices. Our experimental observations are in good agreement with the numerical simulations. These results open the door for investigating how anisotropy and nonlocality influence the formations of lattice structure and lattice solitons, such as elliptical gap solitons and vortex solitons. Since the lattice structure can be tuned at ease by altering the relative orientations among the c-axis, bias field, and lattice beam, we expect this can lead to a novel approach for band-gap engineering as well as diffraction and refraction management.

Fig. 4. Experimental observations of elliptical discrete solitons. Results in (a)–(c) correspond to the cases of θe = θc = π/4; θe = 3π/4, θc = π/4; and θe = 5π/8, θc = π/8, respectively, where the left (3D intensity plots) and middle (2D transverse patterns) columns are experimental results, and the right column is from numerical simulation for comparison. The top and bottom rows in (a)–(c) show the linear and nonlinear output beam patterns, respectively.

Acknowledgments

This work was supported by the Youth for NPU Teachers Scientific and Technological Innovation Foundation, the NPU Foundation for Fundamental Research, the Doctorate Foundation of NPU, the 973 program, 111 project, NSFC, PCSIRT, NSF and AFOSR.

References and links

1.

For a review, see D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003). [CrossRef] [PubMed]

2.

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E 66, 046602 (2002). [CrossRef]

3.

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147–150 (2003). [CrossRef] [PubMed]

4.

D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Królikowski, “Spatial solitons in optically induced gratings” Opt. Lett. 28, 710–712 (2003). [CrossRef] [PubMed]

5.

H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, “Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices,” Phys. Rev. Lett. 92, 123902 (2004). [CrossRef] [PubMed]

6.

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides. “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett.90, 023902 (2003). [CrossRef] [PubMed]

7.

A. A. Zozulya and D. Z. Anderson, “Propagation of an optical beam in a photorefractive medium in the presence of a photogalvanic nonlinearity or an externally applied electric field,” Phys. Rev. A. 51, 1520–1531 (1995). [CrossRef] [PubMed]

8.

A. A. Zozulya, D. Z. Anderson, A. V. Mamaev, and M. Saffman, “Self-focusing and soliton formation in media with anisotropic nonlocal material response,” Europhys. Lett. 36, 419–424 (1996). [CrossRef]

9.

P. Zhang, J. Zhao, C. Lou, X. Tan, Y. Gao, Q. Liu, D. Yang, J. Xu, and Z. Chen, “Elliptical solitons in nonconventionally biased photorefractive crystals,” Opt. Express 15, 536–544 (2007). [CrossRef] [PubMed]

10.

E. D. Eugenieva, D. N. Christodoulides, and M. Segev, “Elliptic incoherent solitons in saturable nonlinear media,” Opt. Lett. 25, 972–974 (2000). [CrossRef]

11.

O. Katz, T. Carmon, T. Schwartz, M. Segev, and D. N. Christodoulides, “Observation of elliptic incoherent spatial solitons,” Opt. Lett. 29, 1248–1250 (2004). [CrossRef] [PubMed]

12.

J. Hudock, N. K. Efremidis, and D. N. Christodoulides, “Anisotropic diffraction and elliptic discrete solitons in two-dimensional waveguide arrays,” Opt. Lett. 29, 268–270 (2004). [CrossRef] [PubMed]

13.

F. Ye, L. Dong, J. Wang, T. Cai, and Y. Li, “Discrete elliptic solitons in two-dimensional waveguide arrays,” Chin. Opt. Lett. 3, 227–229 (2005).

14.

P. G. Kevrekidis, D. J. Frantzeskakis, R. Carretero-González, B. A. Malomed, and A. R. Bishop “Discrete solitons and vortices on anisotropic lattices,” Phys. Rev. E72, 046613 (2005). [CrossRef]

15.

B. Terhalle, A. S. Desyatnikov, C. Bersch, D. Träger, L. Tang, J. Imbrock, Y. S. Kivshar, and C. Denz, “Anisotropic photonic lattices and discrete solitons in photorefractive media,” Appl. Phys. B 86, 399–405 (2007). [CrossRef]

16.

P. Zhang, S. Liu, J. Zhao, C. Lou, J. Xu, and Z. Chen, “Optically induced transition between discrete and gap solitons in a nonconventionally biased photorefractive crystal,” to appear in Opt. Lett. (2008). [CrossRef] [PubMed]

17.

P. Zhang, Y. Ma, J. Zhao, D. Yang, and H. Xu, “One-dimensional spatial dark soliton-induced channel waveguides in lithium niobate crystal,” Appl. Opt. 45, 2273–2278 (2006). [CrossRef] [PubMed]

18.

V. I. Petviashvili, “Equation of an extraordinary soliton,” Sov. J. Plasma Phys. 2, 257–258 (1976).

OCIS Codes
(190.4420) Nonlinear optics : Nonlinear optics, transverse effects in
(190.6135) Nonlinear optics : Spatial solitons

ToC Category:
Nonlinear Optics

History
Original Manuscript: January 25, 2008
Revised Manuscript: February 27, 2008
Manuscript Accepted: February 27, 2008
Published: March 10, 2008

Citation
Peng Zhang, Jianlin Zhao, Fajun Xiao, Cibo Lou, Jingjun Xu, and Zhigang Chen, "Elliptical discrete solitons supported by enhanced photorefractive anisotropy," Opt. Express 16, 3865-3870 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-6-3865


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References

  1. For a review, see D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817-823 (2003). [CrossRef] [PubMed]
  2. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, "Discrete solitons in photorefractive optically induced photonic lattices," Phys. Rev. E 66, 046602 (2002). [CrossRef]
  3. J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices," Nature 422, 147-150 (2003). [CrossRef] [PubMed]
  4. D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Królikowski, "Spatial solitons in optically induced gratings," Opt. Lett. 28, 710-712 (2003). [CrossRef] [PubMed]
  5. H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, "Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices," Phys. Rev. Lett. 92, 123902 (2004). [CrossRef] [PubMed]
  6. J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides. “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90, 023902 (2003). [CrossRef] [PubMed]
  7. A. A. Zozulya and D. Z. Anderson, "Propagation of an optical beam in a photorefractive medium in the presence of a photogalvanic nonlinearity or an externally applied electric field," Phys. Rev. A 51, 1520-1531 (1995). [CrossRef] [PubMed]
  8. A. A. Zozulya, D. Z. Anderson, A. V. Mamaev, and M. Saffman, "Self-focusing and soliton formation in media with anisotropic nonlocal material response," Europhys. Lett. 36, 419-424 (1996). [CrossRef]
  9. P. Zhang, J. Zhao, C. Lou, X. Tan, Y. Gao, Q. Liu, D. Yang, J. Xu, and Z. Chen, "Elliptical solitons in nonconventionally biased photorefractive crystals," Opt. Express 15, 536-544 (2007). [CrossRef] [PubMed]
  10. E. D. Eugenieva, D. N. Christodoulides, and M. Segev, "Elliptic incoherent solitons in saturable nonlinear media," Opt. Lett. 25, 972-974 (2000). [CrossRef]
  11. O. Katz, T. Carmon, T. Schwartz, M. Segev, and D. N. Christodoulides, "Observation of elliptic incoherent spatial solitons," Opt. Lett. 29, 1248-1250 (2004). [CrossRef] [PubMed]
  12. J. Hudock, N. K. Efremidis, and D. N. Christodoulides, "Anisotropic diffraction and elliptic discrete solitons in two-dimensional waveguide arrays," Opt. Lett. 29, 268-270 (2004). [CrossRef] [PubMed]
  13. F. Ye, L. Dong, J. Wang, T. Cai, and Y. Li, "Discrete elliptic solitons in two-dimensional waveguide arrays," Chin. Opt. Lett. 3, 227-229 (2005).
  14. P. G. Kevrekidis, D. J. Frantzeskakis, R. Carretero-González, B. A. Malomed, and A. R. Bishop "Discrete solitons and vortices on anisotropic lattices," Phys. Rev. E 72, 046613 (2005). [CrossRef]
  15. B. Terhalle, A. S. Desyatnikov, C. Bersch, D. Träger, L. Tang, J. Imbrock, Y. S. Kivshar, and C. Denz, "Anisotropic photonic lattices and discrete solitons in photorefractive media," Appl. Phys. B 86, 399-405 (2007). [CrossRef]
  16. P. Zhang, S. Liu, J. Zhao, C. Lou, J. Xu, and Z. Chen, "Optically induced transition between discrete and gap solitons in a nonconventionally biased photorefractive crystal," to appear in Opt. Lett. (2008). [CrossRef] [PubMed]
  17. P. Zhang, Y. Ma, J. Zhao, D. Yang, and H. Xu, "One-dimensional spatial dark soliton-induced channel waveguides in lithium niobate crystal," Appl. Opt. 45, 2273-2278 (2006). [CrossRef] [PubMed]
  18. V. I. Petviashvili, "Equation of an extraordinary soliton," Sov. J. Plasma Phys. 2, 257-258 (1976).

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